In recent years, as the number of automobiles has increased countermeasures against air pollution have been included as part of countermeasures against public hazards from a viewpoint of environmental contamination. At the same time, countermeasures against fuel consumption have been considered from a viewpoint of energy saving. As one approach for resolving the air pollution problem, a tri-system catalyst has been frequently used. The tri-system catalyst exhibits its highest catalytic action when the air to fuel ratio of the air/fuel mixture is equal to the stoichiometric air to fuel ratio. In order to assure that the tri-system catalyst acts effectively, the air to fuel ratio has to be continuously controlled within a narrow range around the stoichiometric air to fuel ratio while the engine rotation speed of the automobile changes over a very wide range from 600 to 6000 r.p.m. and it rapidly varies. Accordingly, an exhaust gas sensor has been used to sense the exhaust gas condition.
In a system for controlling the air to fuel ratio of the engine, an O.sub.2 sensor for sensing the oxygen content in the exhaust gas has been used and a detection signal of the O.sub.2 sensor has been fed back for control. This air to fuel ratio control system provides a relatively stable control when the engine rotation speed is constant under certain conditions, that is, when the automobile is running at a substantially constant speed. However, as is well known, the engine is operated in various operation modes such as warming up, idling, acceleration and deceleration modes and the operation mode rapidly changes from one to the other depending on the environmental conditions. Accordingly, if the air to fuel ratio is disturbed by the rapid change of the operation mode of the engine, the disturbance may be sensed by an O.sub.2 sensor coupled to the exhaust pipe. Since the time required to sense the disturbance after it has occurred is equal to a sum of the delay time of engine suction and gas exhaust, a waste time L for the exhaust gas to flow through the exhaust pipe and reach the O.sub.2 sensor and a time T from the arrival of the exhaust gas change due to the disturbance to the O.sub. 2 sensor to the generation of an electromotive force by the O.sub.2 sensor (i.e., the time constant of the O.sub.2 sensor), the feedback control by the simple O.sub.2 sensor cannot follow the rapidly changing operation mode.
Accordingly, in order to compensate for the delay of the detection of the exhaust gas by the O.sub.2 sensor and improve the stability of the control, it has been proposed to convert the output waveform of the O.sub.2 sensor to a waveform including a proportional component and an integration component to effect a proportional-integral control. This approach, however, is not sufficient to precisely follow the complex operation mode of the engine.